I like Ilford Multigrade. It's convenient as a liquid, it's cheap, and you can get it in 5 litre cans, which is nice if you use a lot.

Go with tongs for handling prints - I've alays been happy with the plastic patterson ones, although you do need to squeeze them fairly hard to keep a good grip.

5x7 VC paper should be fine, provided that you have the VC filters (or a VC or colour head) to go with them.

I think you should be OK as regards exposure if you are in a well ventilated area. The old printers used to rub prints in the developer with their bare hands to bring areas out further. While this isn't recommended, it does show that the basic b&w chemistry is fairly benign.
Keep a towel handy, and you should be fine.

Good luck, and I'm sure you'll really enjoy the darkroom work. It's really magic seeing your first print appear on the paper!